Problem 2

Let

A

be one of the two distinct intersection points of unequal coplanar circles

C_1,C_2

with centers

O_1,O_2

. One common tangent touches them at

P_1,P_2

, and the other at

Q_1,Q_2

. Let

M_1,M_2

be the midpoints of

P_1Q_1,P_2Q_2

. Prove that

\angle O_1AO_2=\angle M_1AM_2

Step 4 of 5: Compare the two centers

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\angle O_1AM_1=\angle O_2AM_2

Detailed analysis

Because $S,O_1,O_2$ are collinear, the rays $SO_1$ and $SO_2$ represent the same line. Therefore the two angles in Step 3 have equal directed values: $\angle O_1AM_1=\angle O_1SA=\angle O_2SA=\angle O_2AM_2$ .